PHYSICAL REVIEW B 93, 155422 (2016)

Radiative and nonradiative exciton energy transfer in monolayers of two-dimensional group-VItransition metal dichalcogenides

Christina Manolatou, Haining Wang, Weimin Chan, Sandip Tiwari, and Farhan Rana*

School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA(Received 25 January 2016; revised manuscript received 9 March 2016; published 18 April 2016)

We present results on the rates of interlayer energy transfer between excitons in monolayers of two-dimensionalgroup-VI transition metal dichalcogenides (TMDs). We consider both radiative (mediated by real photons) andnonradiative (mediated by virtual photons) mechanisms of energy transfer using a unified Green’s functionapproach that takes into account modification of the exciton energy dispersions as a result of interactions. Thelarge optical oscillator strengths associated with excitons in TMDs result in very fast energy transfer rates. Theenergy transfer times depend on the exciton momentum, exciton linewidth, and the interlayer separation andcan range from values less than 100 femtoseconds to more than tens of picoseconds. Whereas inside the lightcone the energy transfer rates of longitudinal and transverse excitons are comparable, outside the light cone theenergy transfer rates of longitudinal excitons far exceed those of transverse excitons. Average energy transfertimes for a thermal ensemble of longitudinal and transverse excitons is temperature dependent and can be smallerthan a picosecond at room temperature for interlayer separations smaller than 10 nm. Energy transfer times oflocalized excitons range from values less than a picosecond to several tens of picoseconds. When the excitonscattering and dephasing rates are small, energy transfer dynamics exhibit coherent oscillations. Our results showthat electromagnetic interlayer energy transfer can be an efficient mechanism for energy exchange between TMDmonolayers.

DOI: 10.1103/PhysRevB.93.155422

I. INTRODUCTION

The optoelectronic properties of 2D transition metaldichalcogenide (TMD) monolayers are dominated by excitons[1–3]. Distinguishing features of the excitons in 2D metaldichalcogenides are the large exciton binding energies andthe strong exciton-photon interactions [1–7]. Recently, excitonpolaritons have been also studied experimentally and theoret-ically in these materials [7–11]. The strong exciton-photoncoupling results in spontaneous emission radiative lifetimesin the hundreds of femtoseconds range [7,12–14]. The strongexciton-photon coupling suggests that the rates for interlayerenergy transfer between excitons in parallel TMD monolayerswould also be fast.

In electronically coupled 2D TMD monolayers, ultrafastenergy transfer via interlayer charge transfer has been observed[15–17]. In this paper, we study the rate of transfer of energybetween excitons in parallel (electronically decoupled) TMDmonolayers as a result of electromagnetic coupling. Themechanism for this energy exchange could be both radiative(mediated by propagating photons for exciton states withinthe light cone [7]) or nonradiative (mediated by evanescentphotons that are bound to the exciton states outside the lightcone in an isolated TMD layer as exciton polaritons but canmediate energy exchange between two TMD layers if thetwo TMD layers are close enough). The latter mechanismis the same as the well known Forster resonance energy trans-fer (FRET) mechanism due to dipole-dipole coupling [18].However, the use of the standard FRET dipole-dipole energyexchange formulas in the present case gives erroneous resultssince it ignores the retarded nature of the exciton interlayerinteraction [19,20]. The quantum electrodynamic Green’s

*fr37@cornell.edu

function approach used here treats both these mechanismson equal footing while taking into account the corrections tothe longitudinal and the transverse exciton energy dispersionrelations due to coupling with the radiation.

The energy transfer times depend on the exciton momen-tum, exciton intralayer scattering rates, and the interlayer sepa-ration. Our results show that the large exciton optical oscillatorstrengths in TMD monolayers result in energy transfer timesshorter than 100 fs for longitudinal excitons for interlayerspacings smaller than 10 nm. Average energy transfer timesfor a thermal ensemble of longitudinal and transverse excitonscan be smaller than a picosecond. We also consider localizedexcitons and find that localized longitudinal excitons canalso have energy transfer times shorter than a picosecondfor interlayer spacings smaller than 10 nm. If the excitonscattering rates are fast, energy transfer involves a simple decayof energy from one layer to the other. If the exciton scatteringrates are slow, energy transfer dynamics can exhibit coherentoscillations. Conditions for observing such oscillations arediscussed in this paper. Our results show that electromagneticinterlayer energy transfer can be an efficient mechanismfor energy exchange between electronically uncoupled TMDmonolayers and can even compete with the interlayer chargetransfer mechanism in the case of electronically coupled TMDlayers.

Section II discusses exciton states in TMDs, longitudinaland transverse excitons in TMDs, and the Hamiltonian termsthat describe exciton-photon interaction in TMDs. Sections IIIand IV present the main results of this paper on the energytransfer rates between two TMD layers. Section V presentsnumerical results for energy transfer rates between two MoS2

layers. Section VI considers the case where the excitonintralayer scattering and dephasing rates are small and energytransfer dynamics exhibit coherent oscillations. Sections VIIand VIII discuss some special cases, including that of

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MANOLATOU, WANG, CHAN, TIWARI, AND RANA PHYSICAL REVIEW B 93, 155422 (2016)

localized excitons and energy transfer between excitons andfree electron-hole pairs in TMD heterolayers. The Appendixdiscusses exciton self-energies and energy dispersions in TMDmonolayers in the presence of exciton-photon interaction.Although most of the numerical results presented in this paperare for MoS2 monolayers, the analysis and the results presentedare expected to be relevant to all TMDs and are expectedto be useful in realizing new kinds of metal dichalcogenideoptoelectronic devices based on energy transfer.

II. PRELIMINARIES

A. Energy bands in TMDs

The conduction and valence bands in monolayer TMDsnear the K and K ′ points in the Brillouin zone are describedby the Hamiltonian [21],[

�/2 �vk−�vk+ −�/2 + λτσ

]. (1)

Here, � is related to the material band gap, σ = ±1 stands forthe electron spin, τ = ±1 stands for the K and K ′ valleys, 2λ

is the splitting of the valence band due to spin-orbit coupling,k± = τkx ± iky , and the velocity parameter v is related to thecoupling between the orbitals on neighboring M atoms. Fromdensity functional theories [22,23], v ≈ 5 − 6 × 105 m/s. Thewave vectors are measured from the K(K ′) points. The d-orbital basis used in writing the above Hamiltonian are |dz2〉and (|dx2−y2〉 + iτ |dxy〉)/

√2 [21]. We will use the symbol s for

the combined valley (τ ) and spin (σ ) degrees of freedom. Theintravalley momentum matrix element between the conductionand valence band states near K(K ′) points follows from theabove Hamiltonian,

�Pvc,s(�k′,�k)

= 〈vv,�k′,s | �P |vc,�k,s〉= mov [(x + iτ y) cos(θ�k′,s/2) cos(θ�k,s/2)e−iτ (φ�k′ +φ�k )/2

−(x − iτ y) sin(θ�k′,s/2) sin(θ�k,s/2)eiτ (φ�k′ +φ�k )/2]. (2)

Here, mo is the free electron mass, φ�k is the phase of thewavevector �k, and

cos(θ�k,s) = �s

2√

(�s/2)2 + (�vk)2. (3)

Near the band extrema, cos(θ�k,s) → 1 and �Pvc,s(�k′,�k) →mov (x + iτ y)e−iτ (φ�k′+φ�k )/2.

B. Exciton states in TMDs

Exciton states in TMDs have been discussed in detail inseveral published works [4–7,24]. We assume an undopedintrinsic TMD layer. The creation operator for an exciton statewith center-of-mass momentum �Q is defined as [7,24],

B†�Q,s,α

= 1√A

∑�k

ψ �Q,s,α(�k)c†�k+λe�Q,s

b�k−λh�Q,s. (4)

c�k,s and b�k,s are the destruction operators for the conduction

band and valence band electron states, respectively, ψ �Q,s,α(�k)

is the exciton relative wave function, and A is the area ofthe TMD monolayer. λe = me/mex , λe = mh/mex , mex =me + mh, where me, mh, and mex are the effective masses ofelectrons, holes, and excitons, respectively [7]. The subscriptα corresponds to the different bound exciton levels. Onlyoptically active exciton levels (whose relative wave functionhave a nonzero amplitude at zero relative distance) will beconsidered here. The exciton Hamiltonian can be writtenapproximately as,

Hex =∑�Q,α,s

Eex,α( �Q) B†�Q,s,α

B �Q,s,α. (5)

Here, Eex,α( �Q) is the exciton energy measured with respectto the ground state consisting of a filled valence band and anempty conduction band.

C. Exciton-photon interaction in TMDs: Exciton-polaritons

The quantized radiation field is [25,26]

�A(�r) =∑�q,j

√�

2εoωq

[n�q,j a�q,j + n∗−�q,j a

†−�q,j

]ei �q.�r√

V

=∑�q,j

√�

2εoωq

�R�q,j

ei �q.�r√

V. (6)

Here, n�q,j for j = 1,2 are the field polarization vectors anda�q,j is the field destruction operator for a mode with wavevector �q and frequency ωq . We assume a TMD monolayeroccupying the x-y plane located at z along the z axis. Theinteraction between the electrons and photons is given by theHamiltonian,

Hint =∑j,s

H+j,s + H−

j,s (7)

where,

H−j,s = e

mo

∑�q,�k‖,α

√�

2V Aεoωq

e−iqzz

× �R−�q,j · �Pvc,s(�k‖ − λh �q‖,�k‖ + λe �q‖)ψ�q‖,s,α(�k‖)B�q‖,s,α

(8)

and H+j,s = [H−

j,s]†. We have expressed the field wave vector

�q in terms of the in-plane component �q‖ and the out-of-planecomponent qzz.

1. Transformation to decoupled longitudinaland transverse exciton basis

Excitons belonging to a particular valley (K or K ′) arenot the eigenstates in the presence of long-range dipole-dipoleinteractions [7]. It is therefore convenient to switch excitonsbasis. Transverse and longitudinal exciton states, which aresuperpositions of exciton states belonging to the two valleys,are a much better choice. Transverse exciton exciton statescouple with only TE polarized (or s-polarized) radiation andlongitudinal exciton states couple with only TM polarized (orp-polarized) radiation [7]. We define creation operators for

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transverse (T ) and longitudinal (L) exciton states as follows,

B†�q‖,L,α

= 1√2

[e−iφ�q‖ B†�q‖,τ=1,α

+ eiφ�q‖ B

†�q‖,τ=−1,α

]

B†�q‖,T ,α

= i√2

[e−iφ�q‖ B†�q‖,τ=1,α

− eiφ�q‖ B

†�q‖,τ=−1,α

]. (9)

We also define exciton operator C�q‖,L/T ,α and photon operatorR�q,L/T as

C�q‖,L/T ,α = B�q‖,L/T ,α + B†−�q‖,L/T ,α

R�q,L/T = a�q,L/T + a†−�q,L/T

. (10)

The optical matrix element can be expressed in terms ofχex,α(�r,�q‖),

χex,α(�r,�q‖)

=∫

d2�k‖(2π )2

�Pvc,s(�k‖ − λh �q‖,�k‖ + λe �q‖).xψ�q‖,s,α(�k‖)ei�k‖.�r .

(11)

Using these definitions, the exciton-photon Hamiltonian be-comes

Hint = HL + HT (12)

where,

HL = e

mo

∑�q,α

√A�

V εoωq

e−iqzz

×χex,α(�r = 0,�q‖)|qz|q

R−�q,L C�q‖,L,α

HT = e

mo

∑�q,α

√A�

V εoωq

e−iqzz

×χex,α(�r = 0,�q‖)R−�q,T C�q‖,T ,α. (13)

The basis transformation decouples the longitudinal andtransverse exciton polaritons. In what follows we will assumethat χex,α(0,�q‖) is real. If it had a phase, it could be absorbedin φ�q‖ used to define the operators B�q‖,L/T ,α above.

2. Accounting for the twist angle of the TMD monolayers

The two TMD monolayers shown in Fig. 1 can betwisted with respect to each other. Suppose the second TMDmonolayer is twisted at an angle θo with respect to the firstlayer. The optical matrix elements of the second layer acquireconstant phase factors which can be absorbed in the definitionsof the longitudinal and transverse exciton states given inEq. (9) by letting φ�q‖ go to φ�q‖ − θo in Eq. (9). The resultingexciton-photon Hamiltonian for the twisted layer will thenhave the same form as given in Eq. (13). The exciton energytransfer rates will therefore not depend on the twist anglebetween the two monolayers. This result is not surprisingsince exciton optical absorption in a TMD monolayer doesnot depend on the direction of the in-plane light polarization.

3. The quadratic part of the interaction Hamiltonian

The interaction Hamiltonian used above ignores thequadratic vector potential terms [7,27]. Following Girlanda

FIG. 1. Exciton energy transfer between two parallel TMD layersseparated by a distance d along the z axis. A MX2 TMD monolayerconsists of three layers of atoms: a layer of transition metal M atomssandwiched on both sides with layers of X atoms.

et al. [27], the form of these extra term is found to be,

H ′int = H ′

L + H ′T

H ′L =

∑�Q,α

Eex,α( �Q)

×⎡⎣ e

mo

∑�q

δ�q‖, �Q

√A�

V εoωq

eiqzz|χex(0,�q‖)|Eex,α(�q‖)

|qz|q

R�q,L

⎤⎦

×⎡⎣ e

mo

∑�k

δ�k‖, �Q

√A�

V εoωk

e−ikzz|χex(0,�k‖)|Eex,α(�k‖)

|kz|k

R−�k,L

⎤⎦

H ′T =

∑�Q,α

Eex,α( �Q)

×⎡⎣ e

mo

∑�q

δ�q‖, �Q

√A�

V εoωq

eiqzz|χex(0,�q‖)|Eex,α(�q‖)

R�q,T

⎤⎦

×⎡⎣ e

mo

∑�k

δ�k‖, �Q

√A�

V εoωk

e−ikzz|χex(0,�k‖)|Eex,α(�k‖)

R−�k,T

⎤⎦.

(14)

These terms must be added to the interaction Hamiltonianwhen expressions for the exciton self-energies are calculated[7].

D. Exciton spectral density functions

We define the exciton Green’s function G<�Q,L/T ,α

(t − t ′) asfollows [28],

G<�Q,L/T ,α

(t − t ′) = − i

�〈B†

�Q,L/T ,α(t ′)B �Q,L/T ,α(t)〉. (15)

The angular brackets indicate averaging with respect to anensemble of excitons. In the frequency domain,

G<�Q,L/T ,α

(ω) = − i

�A �Q,L/T ,α(ω)nB

�Q,L/T ,α(ω). (16)

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MANOLATOU, WANG, CHAN, TIWARI, AND RANA PHYSICAL REVIEW B 93, 155422 (2016)

Here, A �Q,L/T ,α(ω) is the exciton spectral density functionand nB

�Q,L/T ,α(ω) is the exciton occupation factor and equals

the Bose-Einstein factor in thermal equilibrium. Most otherexciton Green’s functions can be obtained from the spec-tral density function [28], which incorporates effects dueto exciton-photon interaction as well as intra-layer excitonscattering. The average exciton number is

〈n �Q,L/T ,α〉 = 〈B†�Q,L/T ,α

B �Q,L/T ,α〉

=∫

dω

2πi� G<

�Q,L/T ,α(ω). (17)

The spectral density functions satisfy the sum rule,∫dω

2πA �Q,L/T ,α(ω) = 1. (18)

The energy dispersions Eex,L/T ,α( �Q) and the spectral densityfunctions of the transverse and longitudinal excitons in thepresence of exciton-photon interaction can be found fromthe corresponding retarded Green’s functions [7], as shownin the Appendix. A convenient phenomenological choice forA �Q,L/T ,α(ω) is a Lorentzian,

A �Q,L/T ,α(ω) = 2�� �Q,L/T ,α

(�ω − Eex,L/T ,α( �Q))2 + �2�Q,L/T ,α

. (19)

The FWHM exciton linewidth is 2� �Q,L/T ,α . The one instancewhere the above simple Lorentzian form does not work wellis in the case of the transverse excitons right when the spectralweight shifts between the two branches of the polaritondispersion when moving from inside the light cone to outsidethe light cone (see the Appendix).

III. RATES OF EXCITON ENERGY TRANSFER

We consider two parallel (not necessarily identical) elec-tronically decoupled (but electromagnetically coupled) TMDmonolayers, labeled a and b, located at z = 0 and z = d,respectively, as shown in Fig. 1. We assume that the excitonintralayer scattering and dephasing rates are fast so that energytransfer dynamics can be described as a simple decay. Thecase where exciton scattering and dephasing rates are slowand coherent dynamics are important is discussed later in thispaper in Sec. VI. We calculate the average of the rate of changeof the number of excitons with in-plane momentum �Q in layera as a result of electromagnetic coupling to layer b. The desiredHeisenberg operator is

n �Q,L/T ,a,α =d B

†�Q,L/T ,a,α

B �Q,L/T ,a,α

dt

= − i

�[n �Q,L/T ,a,α,Hint + H ′

int]. (20)

Note that the layer index (a or b) is added to the subscripts.We calculate the average of the operator n �Q,L/T ,a,α using thenonequilibrium Green’s function technique [28],

〈n �Q,L/T ,a,α(t)〉= ⟨

Tc

[e− i

�

∫c (Hint(t ′)+H ′

int(t′))dt ′ n �Q,L/T ,a,α(t)

]⟩. (21)

FIG. 2. Feynman diagrams representing the transfer of energybetween an exciton in layer a and an exciton in layer b by photonexchange. Green’s functions of excitons (straight lines) and photons(wavy lines) are shown in the figure. The subscripts a and b standfor layer a and layer b, respectively. Internal variables, qz and kz, areintegrated over. (Left) Processes involving bare Green’s functions.(Right) Dressed Green’s functions.

Here, Tc stands for operator contour ordering along theKeldysh contour c that runs from time −∞ to +∞ and back[28]. The angled brackets stand for averaging with respectto the initial density matrix at time −∞ [28]. The aboveexpression can be evaluated in terms of Green’s functionsusing standard perturbation techniques [28]. The lowest ordernonzero terms in the perturbative expansion above give therate of decrease of the exciton number due to spontaneousemission into free-space, as discussed by Wang et al. [7].The terms relevant to the present discussion correspond tothe Feynman diagrams depicted in Fig. 2 which representenergy transfer between the excitons in the two layers. Thebare exciton Green’s functions of each layer in Fig. 2 aredressed from, (i) intralayer photon interactions (or intralayerlong-range dipole-dipole interactions), as discussed by Wanget al. [7], and from (ii) intralayer interactions responsible forexciton scattering assuming these interactions are included inthe Hamiltonian.

The final result for the energy transfer rate RE can be writtenin terms of the spectral density functions of the excitons in thetwo layers. For the transverse excitons we get

RE = 〈n �Q,T ,a,α(t)〉

= − 1

�2

∑β

∫dω

2πA �Q,T ,a,α(ω)A �Q,T ,b,β (ω)

×∣∣∣∣ηo

e2

m2o

χex,a,α(0, �Q)χex,b,β (0, �Q)

∣∣∣∣2 |ei

√ω2−Q2c2d/c|2

|ω2 − Q2c2|× [

nB�Q,T ,a,α

(ω) − nB�Q,T ,b,β

(ω)]

(22)

and for the longitudinal excitons we obtain,

RE = 〈n �Q,L,a,α(t)〉

= − 1

�2

∑β

∫dω

2πA �Q,L,a,α(ω)A �Q,L,b,β (ω)

×∣∣∣∣ηo

e2

m2o

χex,a,α(0, �Q)χex,b,β (0, �Q)

∣∣∣∣2

|ei√

ω2−Q2c2d/c|2

× |ω2 − Q2c2|ω4

[nB

�Q,L,a,α(ω) − nB

�Q,L,b,β(ω)

]. (23)

Note that the expressions above are valid for ω > Qc (radia-tive transfer) as well as for ω < Qc (nonradiative transfer)

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RADIATIVE AND NONRADIATIVE EXCITON ENERGY . . . PHYSICAL REVIEW B 93, 155422 (2016)

provided in the latter case the replacement√

ω2 − Q2c2 →i√

Q2c2 − ω2 is made. The above expressions represent themain results of this paper.

IV. REMARKS ON THE ENERGY TRANSFER RATES

The following points regarding the expressions above needto be noted:

(1) The rate of energy transfer depends on the overlap ofthe spectral density functions of the excitons in the two layers.

(2) Outside the light cone, when ω < Qc, the energytransfer is mediated via evanescent fields and the rate oftransfer decreases exponentially with interlayer separation d

as ∼e−2√

Q2c2−ω2d/c.(3) The energy transfer rates depend inversely on the exci-

ton linewidth (and, therefore, the exciton intralayer scatteringrates) via the exciton spectral density functions.

(4) The energy transfer rates depend on the dielectricconstants (or the refractive indices) of the media surroundingthe two monolayers. In the simple case when the media oneither side of the layers and also in between the layers havethe same dispersionless refractive index n, the impedance ηo

and the speed of light c in the above expressions get replacedby ηo/n and c/n, respectively.

(5) Since the exciton state belonging to one valley canbe considered a superposition of transverse and longitudinalexciton states, its energy transfer rate will be the average ofthe energy transfer rates for the transverse and longitudinalexcitons.

(6) In the static limit, Qc ω, the expressions for the en-ergy transfer rates obtained for the longitudinal excitons havethe same form as those obtained previously for quantum wellexcitons using the static dipole-dipole interaction Hamiltonian[20], which is to be expected.

V. NUMERICAL RESULTS FOR THE EXCITON ENERGYTRANSFER TIMES BETWEEN TWO MoS2 MONOLAYERS

For numerical evaluations of the results, we assumetwo identical and parallel MoS2 layers at a distance d.The electronic and optical parameter values used for MoS2

monolayers are the same as those given previously [6,7,24].We first calculate the energy dispersions for the lowest energy1s longitudinal and transverse excitons (as described in theAppendix), and then use these to compute the energy transferrates. In numerical calculations, a momentum-independentscattering-limited FWHM exciton linewidth of ∼30 meV isassumed. Figure 3 shows the calculated energy transfer timesfor both longitudinal and transverse excitons as a functionof the exciton in-plane momentum Q for different values ofthe interlayer separation d (d = 5, 10, 25, and 50 nm). Thevalue of the momentum Qo, defined by �Qoc = Eex,1s(Qo), is∼9.6 1/μm (see the Appendix). For Q � Qo (inside the lightcone), the energy transfer is via the radiative mechanism andthe energy transfer time is almost independent of Q. WhenQ ∼ Qo, the cusps in the energy transfer times follow thetrends in the radiative lifetimes and the optical conductivities oflongitudinal and transverse excitons (see the Appendix). WhenQ > Qo (outside the light cone), the energy transfer is via the

10−1

100

101

102

103

100

102

104

106

Wavevector Q (μm−1)

En

erg

y T

ran

sfer

Tim

e (p

s)

Longitudinal

Transverse

d = 5, 10, 25, 50nm

FIG. 3. The calculated energy transfer times for the lowest energy1s longitudinal (red dashed) and transverse (blue solid) excitons intwo parallel MoS2 monolayers are plotted as a function of the excitonin-plane momentum Q for different values of the interlayer separationd (d = 5, 10, 25, and 50 nm). The value of the momentum Qo, definedby �Qoc = Eex,1s(Qo), is ∼9.6 1/μm (see the Appendix). A FWHMexciton linewidth of ∼30 meV is assumed.

nonradiative mechanism (via evanescent waves). In the caseof the transverse excitons, the energy transfer rate decreases(and the energy transfer time increases) exponentially with theproduct ∼Qd (for large Q). In the case of the longitudinalexcitons, the energy transfer time first decreases with Q

because the in-plane component of the evanescent radiationalso increases with Q, and then the energy transfer timeincreases exponentially with the product ∼Qd (for large Q).Note that for interlayer separations d less than 10 nm, theenergy transfer times for the longitudinal excitons can be in thehundreds of femtoseconds range or even smaller. These resultsshow the efficacy of the exciton energy transfer mechanism inTMD monolayers.

Simple expressions for the energy transfer times τE, �Q,L/T ,α

can be found for two identical TMD layers when Q < Qo

and Q > Qo (i.e., away from Q = Qo) assuming Lorentzianspectral density functions,

1

τE, �Q,T ,α

≈ �

2γ

(2ηo

e2

m2o

|χex,α(0, �Q)|2)2

e2i√

f �Q,T ,αd/�c

|f �Q,T ,α| (24)

1

τE, �Q,L,α

≈ �

2γ

(2ηo

e2

m2o

|χex,α(0, �Q)|2)2

× e2i√

f �Q,L,αd/�c|f �Q,L,α|

E4ex,L,α( �Q)

(25)

where γ = (� �Q,L/T ,a,α + � �Q,L/T ,b,α), f �Q,L/T ,α =(E2

ex,L/T ,α( �Q) − �2Q2c2), and

√f �Q,L/T ,α = i

√|f �Q,L/T ,α|

when f �Q,L/T ,α < 0 outside the light cone. For Q < Qo

(radiative energy transfer) the above expressions can be

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0 100 200 30010

−1

100

101

102

103

Temperature (K)

En

erg

y T

ran

sfer

Tim

e (p

s)

d = 5, 10, 25, and 50 nm

FWHM = 30 meV

FIG. 4. The calculated average energy transfer times for a thermalensemble of (1s) longitudinal and transverse excitons in two parallelMoS2 monolayers are plotted as a function of the exciton temperaturefor different for different values of the interlayer separation d (d = 5,10, 25, and 50 nm). The exciton FWHM linewidth is assumed to be30 meV.

written as,

1

τE, �Q,L/T ,α

≈ �

2γ

(1

τsp, �Q,L/T ,α

)2

. (26)

Here, τsp, �Q,L/T ,α is the spontaneous emission radiative lifetimeof excitons in a single TMD layer [7] (see the Appendix). In thecase of two MoS2 monolayers, τsp, �Q≈0,L/T ,1s is around 200 fs(for 1s excitons) [7]. Assuming FWHM exciton linewidths of∼30 meV [6,29] and ∼10 meV, the radiative energy transfertimes are ∼3.6 ps and ∼1.2 ps, respectively. Although thesetimes seem short, the radiative energy transfer process willnot be very efficient (efficiency less than ∼15%) because thespontaneous emission time is also very short and only a smallfraction of the photons emitted from one layer get absorbed bythe other layer.

Despite the fast energy transfer rates for the longitudinalexcitons when Q > Qo, their contribution to the energytransfer process is expected to be limited by the relatively smalldensity of the longitudinal excitons in a thermal ensembleof excitons since Eex,L,1s( �Q) > Eex,T ,1s( �Q) for Q > Qo (seethe Appendix). Figure 4 shows the calculated average energytransfer times for a thermal ensemble of (1s) longitudinaland transverse excitons as a function of the exciton temper-ature. The exciton density is assumed to be dilute enoughsuch that the exciton chemical potential is less than the lowestexciton energy level by at least several KT . The excitonFWHM linewidth is assumed to be momentum independentand equal to 30 meV. The results show that when the interlayerseparation is small then as the temperature increases, and thedensity of the longitudinal excitons also increases relativeto the transverse excitons, the average energy transfer timedecreases. However, when the interlayer separation is large,and the energy transfer is by excitons with only small momenta(Q < 1/d), an increase of the temperature results in anincrease of the energy transfer time because the exciton thermal

distribution spills to larger momenta. Since the average excitonenergy transfer time of a thermal ensemble scales with theaverage exciton FWHM linewidth, the average energy transfertime can be shorter than a picosecond for interlayer separationssmaller than 10 nm and average exciton linewidths narrowerthan 10 meV.

VI. COHERENT ENERGY TRANSFER DYNAMICS

In the previous section we assumed that the intralayerexciton scattering and dephasing rates are slow and energytransfer dynamics can be described as a simple decay. Herewe quantify this notion and also discuss coherent interlayerenergy transfer dynamics. First, we evaluate corrections tothe exciton dispersions as a result of interlayer radiative andnonradiative interactions.

We consider two parallel and identical electronically de-coupled (but electromagnetically coupled) TMD monolayers,labeled a and b, located at z = 0 and z = d, respectively,as shown earlier in Fig. 1. The analysis is greatly simplifiedif we define operators for the in-phase (‘+’ exciton) andout-of-phase (‘−’ exciton) excitons in the two layers as follows[30],

B�q‖,L/T ,±,α = B�q‖,L/T ,a,α ± B�q‖,L/T ,b,α√2

. (27)

The ‘+’ and ‘−’ excitons have their dipole moments in-phaseand out-of-phase, respectively. The retarded Green’s functionsand self-energies for the ‘+’ and ‘−’ excitons can be foundusing the methods described in the Appendix,

GR�q‖,L/T ,±,α(ω)

=2Eex,α(�q‖)

[1 − 2 �oR

�q‖,L/T ,±,α(ω)/Eex,α(�q‖)

](�ω)2 − Eex,α(�q‖)2 − 2 (�ω)2

Eex,α(�q‖)�oR�q‖,L/T ,±,α

(ω)

≈ 2Eex,α(�q‖)

(�ω)2 − Eex,α(�q‖)2 − 2 (�ω)2

Eex,α(�q‖)�oR�q‖,L/T ,±,α

(ω). (28)

Here,

�oR�q‖,L/T ,±,α(ω) = �oR

�q‖,L/T ,α(ω)[1 ± ei√

ω2−q2‖ c2d/c]. (29)

�oR�q‖,L/T ,α

(ω) is the retarded self-energy for excitons in a singleTMD layer and its expression was given previously [7] (alsosee the Appendix). The expression above is valid for ω >

q‖c as well as for ω < q‖c provided in the latter case the

replacement√

ω2 − q2‖c

2 → i√

q2‖c

2 − ω2 is made. It is clearfrom the above expression for the self-energy that in the limitd → 0 the out-of-phase ‘−’ excitons do not radiate whereasthe radiative rates of the in-phase ‘+’ excitons are twice asfast as those of excitons in a single TMD layer. The energysplitting between the ‘+’ and ‘−’ excitons due to interlayerinteractions can be estimated as

��q‖,L/T ,α

= Real[2�oR

�q‖,L/T ,α(ω)ei√

ω2−q2‖ c2d/c

]�ω=Eex,L/T ,α (�q‖). (30)

Since an exciton state in any one of the two TMD layerscan be considered a superposition of the in-phase and theout-of-phase exciton states, if the energy splitting ��q‖,L/T ,α

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RADIATIVE AND NONRADIATIVE EXCITON ENERGY . . . PHYSICAL REVIEW B 93, 155422 (2016)

10−1

100

101

102

103−0.02

−0.01

0

0.01

0.02

0.03

Wavevector Q (μm−1)

En

erg

y S

plit

tin

g Δ

(eV

)

Transverse

Longitudinal

d = 5, 10, 25, 50 nm

FIG. 5. Calculated energy splittings between the ‘+’ and ‘−’excitons are plotted for the transverse (blue-solid) and longitudinal(red-dashed) excitons for the lowest energy (1s) exciton state in twosuspended MoS2 monolayers as a function of the in-plane momentumQ for different values of the interlayer spacing d . The value of themomentum Qo, defined by �Qoc = Eex,1s(Qo), is ∼9.6 1/μm.

due to interlayer interactions is much larger than the excitonlinewidth due to intralayer scattering and dephasing thencoherent energy oscillations between exciton states in the twolayers are expected at the frequency ��q‖,L/T ,α/�, and energytransfer between the layers cannot be described as a simpledecay of energy from one layer to the other.

The calculated energy splittings between the ‘+’ and ‘−’excitons are plotted in Fig. 5 for the lowest energy (1s) excitonstate in two suspended MoS2 monolayers as a function of thein-plane momentum Q for different values of the interlayerspacing d (d = 5, 10, 25, and 50 nm). Inside the light cone,the energy splittings are small (less than 1–2 meV) for allvalues of d considered. The energy splittings are smaller thanthe natural linewidth of excitons in a single MoS2 layer dueto radiative decay. Consequently, coherent energy oscillationsare not expected for excitons inside the light cone. Outside thelight cone, the energy splittings for the longitudinal excitonsbecome large reaching values larger than ∼25 meV for d lessthan 5 nm. However, these large energy splittings occur atlarge values of the exciton momenta where exciton intralayerscattering is also expected to be fast and the condition forcoherent oscillations might be difficult to meet. If, outside thelight cone, exciton scattering rates are small then the coherentdynamics of the average exciton layer number 〈n �Q,L/T ,a/b,α〉can be modeled by the following equation similar to that of adamped simple harmonic oscillator,[

d2

dt2+ γ

�

d

dt+

�2�Q,L/T ,α

�2

]〈n �Q,L/T ,a/b,α(t)〉 = 1

2

�2�Q,L/T ,α

�2

(31)

Here, γ = (� �Q,L/T ,a,α + � �Q,L/T ,b,α) is related to the excitonscattering and dephasing rate and the appropriate boundaryconditions are

n �Q,L/T ,a,α(t = 0) = 1

n �Q,L/T ,b,α(t = 0) = 0. (32)

If � �Q,L/T ,α > γ/2, then the above equation predicts dampedoscillations at the frequency � �Q,L/T ,α/�. On the other hand,if � �Q,L/T ,α � γ /2, then the above equation gives a simpleexponential decay of energy from layer a to layer b at therate �2

�Q,L/T ,α/(2�γ ). This latter result is almost exactly what

was obtained earlier in Eqs. (24) and (25). In the absence ofquantitative models or experimental data for exciton scatteringin TMDs it is difficult to say if coherent energy oscillations arepossible in TMDs. In the case of localized excitons, discussednext, momentum spread due to localization also contributes tothe decoherence of the oscillations.

VII. ENERGY TRANSFER RATESFOR LOCALIZED EXCITONS

A. Energy transfer rates for a localized exciton in layer a andfree excitons in layer b

The analysis in the previous sections shows that thelongitudinal excitons with momenta Q in the Qo < Q < 1/d

range have the shortest energy transfer times, but the density ofsuch excitons is relatively small in a thermal ensemble at lowtemperatures thus limiting the average energy transfer rates.Localized excitons, whose wave function is a superpositionof exciton states of different momenta, could overcome theselimitations and exhibit fast energy transfer rates even at lowtemperatures. We consider an initial exciton state in layer a

that is localized in space in a region of size Lc. We assume thatthe exciton wave function for the center of mass coordinate inreal and Fourier spaces is

ψcom( �R) = 1√πL2

c

e−R2/2L2c

ψcom( �Q) =√

4πL2ce

−Q2L2c/2. (33)

A localized exciton state can be constructed from the groundstate |ψo〉, corresponding to a filled valence band and an emptyconduction band, as follows [7],

|ψL/T,a,α〉ex = 1√A

∑�Q

ψcom( �Q)B†�Q,L/T ,a,α

|ψo〉. (34)

Assuming the above localized state as the initial state, with aspectral density function AL/T,a,α(ω) with HWHM linewidthof �L/T,a,α and centered at the energy Eex,L/T ,a,α , the rate ofenergy transfer to free excitons in layer b for the transversecase is found to be,

RE = − 1

�2

∑β

∫d2 �Q(2π )2

|ψcom( �Q)|2

×∫

dω

2πAT,a,α(ω)A �Q,T ,b,β (ω)

×∣∣∣∣ηo

e2

m2o

χex,a,α(0, �Q)χex,b,β (0, �Q)

∣∣∣∣2 |ei

√ω2−Q2c2d/c|2

|ω2 − Q2c2|× [

nBT,a,α(ω) − nB

�Q,T ,b,β(ω)

](35)

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MANOLATOU, WANG, CHAN, TIWARI, AND RANA PHYSICAL REVIEW B 93, 155422 (2016)

and for the longitudinal case we obtain,

RE = − 1

�2

∑β

∫d2 �Q(2π )2

|ψcom( �Q)|2

×∫

dω

2πAL,a,α(ω)A �Q,L,b,β (ω)

×∣∣∣∣ηo

e2

m2o

χex,a,α(0, �Q)χex,b,β (0, �Q)

∣∣∣∣2

|ei√

ω2−Q2c2d/c|2

×|ω2 − Q2c2|ω4

[nB

L,a,α(ω) − nB�Q,L,b,β

(ω)]. (36)

Again note that the expressions above are valid forω < Qc (nonradiative transfer) provided the replacement√

ω2 − Q2c2 → i√

Q2c2 − ω2 is made.Simpler expressions can be obtained in some special

cases. Suppose there exists an exciton state β with momen-tum Q∗ and a corresponding energy E∗ in layer b whichsatisfies the energy conservation relation Eex,L/T ,b,β (Q∗) =Eex,L/T ,b,β (Q = 0) + E∗ = Eex.L/T ,a,α . If the exciton in layera is strongly localized such that the energy spread of the freeexcitons in layer b corresponding to the momentum spreadof the localized exciton in layer a is much greater thanγ = (�L/T,a,α + � �Q,L/T ,b,α), coherent oscillations will not bepossible. If Q∗ < Qo and Q∗ > Qo (i.e., away from Q∗ = Qo)then, assuming Lorentzian spectral density functions, theenergy transfer times can be written as,

1

τE,T ,α

≈ π�

2gex,T ,b,β (E∗)|ψcom(Q∗)|2

×(

2ηo

e2

m2o

|χex,a,α(0, �Q∗)χex,b,β (0, �Q∗)|)2

× e2i√

fT,a,αd/�c

|fT,a,α| (37)

1

τE,L,α

≈ π�

2gex,L,b,β (E∗)|ψcom(Q∗)|2

×(

2ηo

e2

m2o

|χex,a,α(0, �Q∗)χex,b,β (0, �Q∗)|)2

× e2i√

fL,a,αd/�c |fL,a,α|E4

ex,L,a,α

(38)

where fL/T,a,α = (E2ex,L/T ,a,α − �

2Q2c2), and√

fL/T,a,α =i√|fL/T,a,α| when fL/T,a,α < 0 outside the light cone. Here,

gex,L/T ,b,β (E) is the density of states of free excitons in layer b.Note that in this case the energy transfer times do not depend onthe scattering/dephasing rate given by γ . Assuming α = β, theexpressions above differ from the corresponding expressionsfor free excitons, given earlier in Eqs. (24) and (25), bya multiplicative factor of πγgex,L/T ,b,α(E∗)|ψcom(Q∗)|2. Forstrongly localized excitons, this factor is of the order of unityand therefore energy transfer times for strongly localized 1s

excitons in MoS2, when plotted as a function of Q∗, areexpected to be similar to those appearing in Fig. (3).

B. Energy transfer rates for a localized exciton in layer a and alocalized exciton in layer b

We now consider the case in which the final excitonstate in layer b is also localized. The center of mass wavefunctions of the excitons in layer a and b in real space arecentered at the in-plane vectors �ρa and �ρb, respectively, and inmomentum space these wave functions are ψcom,a(�q‖)e−i �q‖. �ρa

and ψcom,b(�q‖)e−i �q‖. �ρb , respectively, where ψcom,a/b(�q‖) are asgiven earlier in Eq. (33). The vector �r connects the center of theexciton states, �r = ( �ρb − �ρa) + dz. The rate of energy transferfor the transverse case is found to be,

RE = − 1

�2

∑β

∫dω

2πAT,a,α(ω)AT,b,β (ω)

×∣∣∣∣∣∫

d3 �q(2π )3

ψ∗com,b(�q‖)ψcom,a(�q‖) ei �q.�r 2c

ω2 − ω2q + iη

× ηo

e2

m2o

χex,a,α(0,�q‖)χex,b,β (0,�q‖)

∣∣∣∣2

× [nB

T,a,α(ω) − nBT,b,β (ω)

](39)

and for the longitudinal case we get,

RE = − 1

�2

∑β

∫dω

2πAL,a,α(ω)AL,b,β (ω)

×∣∣∣∣∣∫

d3 �q(2π )3

ψ∗com,b(�q‖)ψcom,a(�q‖) ei �q.�r 2c

ω2 − ω2q + iη

×(

1 − q2‖

ω2/c2

)ηo

e2

m2o

χex,a,α(0,�q‖)χex,b,β (0,�q‖)

∣∣∣∣∣2

× [nB

L,a,α(ω) − nBL,b,β (ω)

]. (40)

An interesting case is that of extremely localized excitonsfor which Lc � �c/Eex,L/T ,a/b and Lc � d. Assuming wave-vector independent values of χex,a/b,α/β (0,�q‖) and using theresults,∫

d3 �q(2π )3

ei �q.�r 1

ω2 − ω2q + iη

= − ei ωcr

4πrc2

∫d3 �q

(2π )3ei �q.�r

(1 − q2

‖ω2/c2

)ω2 − ω2

q + iη= − ei ω

cr

4πr3ω2

(1 − i

ω

cr

),

(41)

the above expressions for RE , in the limit (ω/c)r � 1, give a1/r2 dependence of the energy transfer rate for the transversecase and a 1/r6 dependence for the longitudinal case. Itis satisfying to note that the former result corresponds tothe classical inverse square law for radiative energy transferand the latter corresponds to the standard Forster’s resultfor nonradiative energy transfer via dipole-dipole interaction[18,31]. In the longitudinal case, if one integrates the energytransfer rate over the in-plane position �ρb of the final excitonstate, then the total energy transfer rate will scale as 1/d4 withthe interlayer separation.

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RADIATIVE AND NONRADIATIVE EXCITON ENERGY . . . PHYSICAL REVIEW B 93, 155422 (2016)

VIII. ENERGY TRANSFER RATES FOR AN EXCITON INLAYER a AND FREE ELECTRON-HOLE PAIRS IN LAYER b

In many cases of practical interest where the optical bandgaps and/or the exciton binding energies in two different TMDmonolayers are very different, the energy transfer can be fromthe excitons in the wider band-gap TMD layer to the freeelectron-hole pairs in the narrower band-gap TMD layer. Forexample, this could be the case in two parallel monolayers ofMoS2 and MoTe2 [32].

We assume that an exciton in layer a with momentum �Qdecays into a free electron-hole pair in layer b. We assumethat the energy emission and absorption remains close to theconduction and valence band edges at K and K ′ valleys so thatthe standard optical selection rules are not violated. Since afree electron-hole pair can be considered an unbound exciton,the expressions for the rate of energy transfer between excitonsgiven in the main text are also valid for energy transfer between

excitons in layer a and free electron-hole pairs in layer b

provided the relative exciton wave function in layer b isassumed to be a plane wave, the exciton energy dispersionin layer b is replaced by that of a free electron-hole pair, andthe summation over the final exciton states (i.e., over β) inlayer b is replaced by a phase space integral over the relativewave vector.

We assume that the free electron-hole pair in layer b isdescribed by the spectral density function A�k, �Q,b(ω) with

HWHM linewidth ��k, �Q,b. Here, �k is the relative momentum

of the electron-hole pair and �Q is the center of massmomentum and the spectral density function is centered at theenergy Eg,b + �

2k2/2mr,b + �2Q2/2mex,b. mr,b is the reduced

electron-hole mass in layer b, mex,b is the exciton mass in layerb, and Eg,b is the band gap of layer b.

For the case of the transverse excitons in layer a

we get,

RE = 〈n �Q,T ,a,α(t)〉 = − 1

�2

∫d2�k

(2π )2

∫dω

2πA �Q,T ,a,α(ω)A�k, �Q,b(ω)

∣∣∣∣ηo

e2

m2o

χex,a,α(0, �Q)χ�k,b( �Q)

∣∣∣∣2 |ei

√ω2−Q2c2d/c|2

|ω2 − Q2c2|× {

nB�Q,T ,a,α

(ω)[fv(�k − λh,b�Q) − fc(�k + λe,b

�Q)] − fc(�k + λe,b�Q)[1 − fv(�k − λh,b

�Q)]}

(42)

and for the longitudinal excitons we obtain,

RE = 〈n �Q,L,a,α(t)〉 = − 1

�2

∫d2�k

(2π )2

∫dω

2πA �Q,L,a,α(ω)A�k, �Q,b(ω)

∣∣∣∣ηo

e2

m2o

χex,a,α(0, �Q)χ�k,b( �Q)

∣∣∣∣2

|ei√

ω2−Q2c2d/c|2

×|ω2 − Q2c2|ω4

{nB

�Q,L,a,α(ω)[fv(�k − λh,b

�Q) − fc(�k+λe,b�Q)]−fc(�k+λe,b

�Q)[1−fv(�k−λh,b�Q)]

}. (43)

fc/v are the layer b conduction and valence band electron occupation factors, and χ�k,b( �Q) is related to the interband momentummatrix element in layer b by the expression,

χ�k,b( �Q) = �Pvc,s(�k − λh�Q,�k + λe

�Q).xei[τφ�k+λe �Q+τφ�k−λh

�Q]. (44)

We define the momentum k∗ and the energy E∗ by the energy conservation relations, Eex,L/T ,a,α(Q) = Eg,b + E∗ + �2Q2/2mex,b

and E∗ = �2(k∗)2/2mr,b. Given the number of possible final states (corresponding to different values of �k) and the fast electron

and hole scattering rates in TMDs, we don’t expect coherent oscillations. Assuming that the narrower band-gap material is in theground state with a full valence band and an empty conduction band, the energy transfer times can be expressed as,

1

τE,T ,α

≈ π�

2gfree,b(E∗)

(2ηo

e2

m2o

|χex,a,α(0, �Q)χ�k∗,b( �Q)|)2

e2i√

f �Q,T ,a,αd/�c

|f �Q,T ,a,α| (45)

1

τE,L,α

≈ π�

2gfree,b(E∗)

(2ηo

e2

m2o

|χex,a,α(0, �Q)χ�k∗,b( �Q)|)2

e2i√

f �Q,L,a,αd/�c|f �Q,L,a,α|

E4ex,L,a,α( �Q)

(46)

where f �Q,L/T ,a,α = (E2ex,L/T ,a,α( �Q) − �

2Q2c2), and√f �Q,L/T ,a,α = i

√|f �Q,L/T ,a,α| when f �Q,L/T ,a,α < 0 outside

the light cone. Here, gfree,b(E) is the joint density of statesfor the creation of free electron-hole pairs (per valley/spin)in layer b. Again note that the energy transfer times do notdepend on the scattering/dephasing rate. The expressionsabove differ from the corresponding expressions for freeexcitons, given earlier in Eqs. (24) and (25), by a multiplicativefactor of πγgfree,b(E∗)|χ�k∗,b( �Q)/χex,a,α(0, �Q)|2. This factor isexpected to be much smaller than unity for most TMD pairs.Still, the energy transfer times for longitudinal excitons in

layer a can range from a picosecond to tens of picoseconds(depending on the initial exciton momentum �Q and themagnitude of the joint density of states for free electron-holepair creation in layer b) for interlayer spacings smaller than10 nm.

As an example, we consider the case of MoS2 and MoTe2

layers. The exciton optical absorption energy in MoS2 is∼1.9 eV and the quasiparticle bandgap Eg,b of MoTe2 is∼1.7 eV. Assuming an exciton FWHM of 30 meV in MoS2

and parameters of MoTe2 as given in the literature [32], thevalue of πγgf ee,b(E∗)|χ�k∗,b( �Q)/χex,a,1s(0, �Q)|2 is found to be

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MANOLATOU, WANG, CHAN, TIWARI, AND RANA PHYSICAL REVIEW B 93, 155422 (2016)

in the 0.07–0.08 range (for different momenta of the initialexciton state in MoS2). Therefore, the energy transfer timesbetween (1s) excitons in MoS2 and free electron-hole pairsin MoTe2 will be approximately 12–14 times those givenin Fig. 3 for the case of 1s excitons in two identical MoS2

layers.

IX. CONCLUDING REMARKS

In this paper we presented results on the energy transferrates between excitons in 2D TMD monolayers. The resultsshow that the energy transfer rates can be very fast. Excitonenergy transfer can potentially be used to design noveloptoelectronic devices with TMD monolayers.

The results presented in this paper can be tested exper-imentally. For example, the energy transfer rate from thelowest energy (1s) excitons in one monolayer to higherenergy excitons or to free electron-hole pairs in another (anddifferent) monolayer can be determined by measuring thephotoluminescence efficiency and/or the photoluminescencelifetime of the lowest energy (1s) excitons as a function ofthe separation between the two monolayers. It might alsobe possible to observe energy transfer more directly. Sinceexcitons in TMD monolayers can be generated (detected)electrically via electron-hole injection (separation) in metal-semiconductor junctions or in-plane pn-junctions [33,34],exciton energy transfer rates between two identical TMDmonolayers can be studied as a function of the interlayerseparation by using one TMD layer as the source and theother as the drain in the energy transfer process. Preliminaryexperimental results using the former scheme have recentlybeen reported by Kozawa et al. [35].

The theory presented in this paper has certain limitationsand care needs to exercised when interpreting the results andcomparing these results with experiments:

(1) The technique used in this paper is valid provided theexciton optical conductivity [6,29] σα(ω) satisfies |ηoσα(ω)| �1, which is typically the case in TMDs [6,29]. If |ηoσα(ω)| � 1,the vacuum field in the vicinity of the TMD layers will getmodified and the expression for the field given in Eq. (6) willno longer be valid. The field would then need to be quantizedin the presence of the TMD layers [36], a task beyond thescope of this paper.

(2) In plotting all the results, a momentum-independentexciton FWHM linewidth was used. Exciton intralayer scatter-ing and dephasing rates are expected to depend on the excitonmomentum. Since the exciton energy transfer rates depend,in most cases, on the exciton scattering rates, a quantitativetheory or experimental data for momentum-dependent excitonscattering rate in TMDs is needed for a better understandingof the dependence of the energy transfer rates on excitonmomenta.

(3) It is well known that radiation emission and absorptionrates are affected by the presence of dielectric interfaces [37].Most experiments on TMD layers are performed with thelayers placed on dielectric substrates. The influence of nearbydielectrics would need to be taken into account in comparingtheory with experiments.

ACKNOWLEDGMENTS

The authors would like to acknowledge helpful discus-sions with Jared Strait, Paul L. McEuen, and Michael G.Spencer, and support from CCMR under NSF Grant No.DMR-1120296, AFOSR-MURI under Grant No. FA9550-09-1-0705, and ONR under Grant No. N00014-12-1-0072.

APPENDIX A: EXCITON SELF-ENERGIESIN A SINGLE TMD LAYER

In this section, we calculate the exciton self-energy in TMDmonolayers. For the sake of simplicity we will assume thatthere is only one significant exciton level labeled by α. Thenoninteracting (bare) retarded Green’s function for the excitonfield is [28]

GoR�Q,L/T ,α

(t−t ′) = − i

�θ (t−t ′)〈[C �Q,L/T ,α(t),C− �Q,L/T ,α(t ′)]〉.

(A1)

The exciton operator C �Q,L/T ,α was defined earlier in Eq. (10).In the Fourier domain the Green’s function is

GoR�Q,L/T ,α

(ω) = 2Eex,α( �Q)

(�ω)2 − Eex,α( �Q)2 + iη. (A2)

The noninteracting (bare) retarded radiation Green’s functionis defined as,

DoR�q,L/T (t − t ′) = − i

�θ (t − t ′) 〈[R�q,L/T (t),R−�q,L/T (t ′)]〉

DoR�q,L/T (ω) = 2�ωq

(�ω)2 − (�ωq)2 + iη. (A3)

First, we dress the radiation Green’s function with term H ′int

in the exciton-photon interaction Hamiltonian that is quadraticin the vector potential (see Sec. II C 3. The resulting dressedGreen’s functions DR

�q,L/T(ω) can be expressed in a form that

will be useful later,

∫dqz

2π

�

εoωq

DR�q,T (ω)

=∫

dqz

2π�

εoωqDoR

�q,T(ω)

1 − 2 e2

m2o

|χex (0,�q‖)|2Eex,α(�q‖)

∫dqz

2π�

εoωqDoR

�q,T(ω)

(A4)

∫dqz

2π

�

εoωq

|qz|2q2

DR�q,L(ω)

=∫

dqz

2π�

εoωq

|qz|2q2 DoR

�q,L(ω)

1 − 2 e2

m2o

|χex (0,�q‖)|2Eex,α (�q‖)

∫dqz

2π�

εoωq

|qz|2q2 DoR

�q,L(ω)

. (A5)

The denominator on the right hand side in the above equationsis generally small and may be neglected. But it plays animportant role when off-shell exciton self-energies are desired,as shown below. The Green’s functions for the radiation fieldare gauge-dependent. It is convenient to choose the temporalgauge in which the scalar potential is set equal to zero (andneed not be taken into account separately) [38]. Henceforth,all results will be given for the temporal gauge. The Dyson

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RADIATIVE AND NONRADIATIVE EXCITON ENERGY . . . PHYSICAL REVIEW B 93, 155422 (2016)

equation for the exciton Green’s function is

GR�q‖,L/T ,α(ω)

= GoR�q‖,L/T ,α(ω)

[1 + �R

�q‖,L/T ,α(ω) · GR�q‖,L/T ,α(ω)

], (A6)

where the retarded self-energies are found to be,

�R�q‖,T ,α(ω) = e2

m2o

|χex(0,�q‖)|2∫

dqz

2π

�

εoωq

DR�q,T (ω)

�R�q‖,L,α(ω) = e2

m2o

|χex(0,�q‖)|2

×∫

dqz

2π

�

εoωq

(ω2 − q2‖c

2)

ω2DR

�q,L(ω). (A7)

The dressed exciton Green’s functions become,

GR�q‖,L/T ,α(ω)

= 2Eex,α(�q‖)

(�ω)2 − Eex,α(�q‖)2 − 2Eex,α(�q‖)�R�q‖,L/T ,α

(ω)

=2Eex,α(�q‖)

[1 − 2 �oR

�q‖,L/T ,α(ω)/Eex,α(�q‖)

](�ω)2 − Eex,α(�q‖)2 − 2 (�ω)2

Eex,α(�q‖)�oR�q‖,L/T ,α

(ω)

≈ 2Eex,α(�q‖)

(�ω)2 − Eex,α(�q‖)2 − 2 (�ω)2

Eex,α(�q‖)�oR�q‖,L/T ,α

(ω). (A8)

Here, �oR�q‖,L/T ,α

(ω) is the same as �R�q‖,L/T ,α

(ω) given inEq. (A7) above except that the bare radiation Green’s functionDoR

�q,L/T(ω) is used in place of DR

�q,L/T(ω). Finally the exciton

spectral density function A�q‖,L/T ,α(ω) can be related to theretarded Green’s function as follows,

−2� Imag{GR

�q‖,L/T ,α(ω)} = A�q‖,L/T ,α(ω) − A�q‖,L/T ,α(−ω).

(A9)

When ω > q‖c (inside the light cone), both the self-energies�R

�q‖,L/T ,α(ω) have a vanishingly small real part and a large

magnitude of the imaginary part. The latter corresponds tothe radiative lifetime of the exciton. The situation is reversedwhen ω < q‖c and then the magnitude of the real part of theself-energy becomes large and the imaginary part vanishes.Therefore, when Eex,α(�q‖) > �q‖c (inside the light cone) onecan ignore corrections to the exciton dispersion, and theradiative lifetime of the exciton can be related to the imaginarypart of the self-energy evaluated on the shell,

1

τsp,�q‖,L/T ,α

= −2

�Imag

{�R

�q‖,L/T ,α(ω)}

�ω=Eex,α ( �Q) (A10)

and, we get [7,10],

1

τsp,�q‖,T ,α

= 2ηoe2

m2o

|χex(0,�q‖)|2 1√E2

ex,α(�q‖) − (�q‖c)2

1

τsp,�q‖,L,α

= 2ηoe2

m2o

|χex(0,�q‖)|2√

E2ex,α(�q‖) − (�q‖c)2

E2ex,α(�q‖)

.

(A11)

When ω < q‖c and the self-energies are real, we get [7,10],

�R�q‖,T ,α(ω) = −ηo�

2e2

m2o

|χex(0,�q‖)|2E2

ex,α(�q‖)

ω2√(q‖c)2 − ω2

�R�q‖,L,α(ω) = ηo�

2e2

m2o

|χex(0,�q‖)|2E2

ex,α(�q‖)

√(q‖c)2 − ω2. (A12)

Note that the corrections obtained from H ′int, which was

quadratic in the vector potential, are important for obtaining thecorrect expressions for the exciton self-energies far off-shell[26].

APPENDIX B: EXCITON ENERGY DISPERSIONSIN A SINGLE TMD LAYER

The exciton self-energy expressions given above can beused to obtain energy dispersions, Eex,L/T ,α(�q‖), for the

FIG. 6. Calculated transverse and longitudinal exciton energydispersions for the lowest energy (1s) exciton state in a suspendedMoS2 monolayer are plotted as a function of the in-plane momentumQ. Also shown are the photon and the bare exciton dispersion relations(dashed lines). (a) The transverse exciton energy dispersion consistsof two curves. The upper curve has all the spectral weight for Q � Qo

and the lower curve gets all the weight when Q Qo, and the spectralweight shifts from the upper curve to the lower curve around Q ≈ Qo.(b) The dispersion for the longitudinal exciton consists of only a singlecurve and the energy dispersion is linear in Q for Q Qo.

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MANOLATOU, WANG, CHAN, TIWARI, AND RANA PHYSICAL REVIEW B 93, 155422 (2016)

longitudinal and transverse excitons. The results are shownin Fig. 6 for the lowest energy (1s) exciton state in asuspended MoS2 monolayer. We define Qo by the equationEex,1s(Qo) = �Qoc. The dispersion for the transverse excitonconsists of two curves (corresponding to the distinct poles ofthe Green’s function). The upper curve has all the spectralweight for Q � Qo and the lower curve gets all the weightwhen Q Qo, and the spectral weight shifts from the upper

curve to the lower curve around Q ≈ Qo. The dispersion forthe longitudinal exciton consists of only a single curve and theenergy dispersion is linear in Q for Q Qo. For Q > Qo,Eex,L,1s( �Q) > Eex,T ,1s( �Q). Therefore, in a thermal ensembleof excitons, transverse exciton density is expected to exceedthe longitudinal exciton density. Note that there are negligiblysmall corrections to the exciton dispersion within the lightcone for both the transverse and longitudinal excitons.

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